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## MA40062: Ordinary differential equations

Owning Department/School: Department of Mathematical Sciences
Credits: 6
Level: Masters UG & PG (FHEQ level 7)
Period: Semester 2
Assessment: EX 100%
Supplementary Assessment: MA40062 Mandatory Extra Work (where allowed by programme regulations)
Requisites: Before taking this unit you must take MA20216 and take MA20218 and take MA20219 and take MA20220 and take MA30041. Students may also find it useful to take MA20221 before taking this unit.
Description: Aims:
To provide an accessible but rigorous treatment of initial-value problems for nonlinear systems of ordinary differential equations, including existence and uniqueness problems, maximally defined solutions and Lyapunov stability theory. Foundations will be laid for advanced studies in dynamical systems and control. The material is also useful in mathematical biology and numerical analysis.

Learning Outcomes:
After taking this unit, students should be able to:
* Demonstrate understanding of existence theory for the initial-value problem, Lipschitz conditions and uniqueness, maximal intervals of existence, local and global flows, limit sets, basic Lyapunov stability theory including LaSalle's invariance principle, asymptotic stability and linearization about equilibria.
* State and prove basic results in the theory of ordinary differential equations including existence and uniqueness of solutions, properties of limit sets, Lyapunov's stability theorem and LaSalle's invariance principle.
* Analyse the stability properties of equilibria in the context of simple examples.

Skills:
Numeracy T/F A
Problem Solving T/F A
Written and Spoken Communication F (solutions to exercise sheets, problem classes)

Content:
Motivating examples from diverse areas. Existence of solutions for the initial-value problem. Maximal intervals of existence. Local Lipschitz conditions and uniqueness of solutions. Local and global flows (dynamical systems). Limit sets and their properties. Stability and asymptotic stability, Lyapunov functions and LaSalle's invariance principle. Linearization about equilibria and asymptotic stability.
Programme availability:

#### MA40062 is Optional on the following programmes:

Department of Computer Science
• USCM-AFM14 : MComp (hons) Computer Science and Mathematics (Full-time) - Year 3
• USCM-AFM14 : MComp (hons) Computer Science and Mathematics (Full-time) - Year 4
• USCM-AKM14 : MComp (hons) Computer Science and Mathematics with Industrial Placement (Full-time with Thick Sandwich Placement) - Year 4
• USCM-AKM14 : MComp (hons) Computer Science and Mathematics with Industrial Placement (Full-time with Thick Sandwich Placement) - Year 5
• USCM-AAM14 : MComp (hons) Computer Science and Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USCM-AAM14 : MComp (hons) Computer Science and Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 5
Department of Mathematical Sciences
• USMA-AFB15 : BSc (hons) Mathematical Sciences (Full-time) - Year 3
• USMA-AKB16 : BSc (hons) Mathematical Sciences (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFB13 : BSc (hons) Mathematics (Full-time) - Year 3
• USMA-AKB14 : BSc (hons) Mathematics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AFB01 : BSc (hons) Mathematics and Statistics (Full-time) - Year 3
• USMA-AKB02 : BSc (hons) Mathematics and Statistics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB02 : BSc (hons) Mathematics and Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AAB14 : BSc (hons) Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFB05 : BSc (hons) Statistics (Full-time) - Year 3
• USMA-AKB06 : BSc (hons) Statistics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB06 : BSc (hons) Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFM14 : MMath Mathematics (Full-time) - Year 3
• USMA-AFM14 : MMath Mathematics (Full-time) - Year 4
• USMA-AAM15 : MMath Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• TSMA-AFM09 : MSc Mathematical Sciences (Full-time)
• TSMA-APM09 : MSc Mathematical Sciences (Part-time)
Department of Physics
• USXX-AFM01 : MSci (hons) Mathematics and Physics (Full-time) - Year 4

Notes:
* This unit catalogue is applicable for the 2013/4 academic year only. Students continuing their studies into 2014/15 and beyond should not assume that this unit will be available in future years in the format displayed here for 2013/14.
* Programmes and units are subject to change at any time, in accordance with normal University procedures.
* Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.